Integral estimates for the trace of symmetric operators

arXiv:1204.0108v2 [math.DG] 12 Aug 2012

INTEGRAL ESTIMATES FOR THE TRACE OF SYMMETRIC OPERATORS. M. BATISTA AND H. MIRANDOLA

Abstract. Let Φ : T M → T M be a positive-semidefinite symmetric operator of class C 1 defined on a complete non-compact manifold M ¯ . In this paper, we given isometrically immersed in a Hadamard space M conditions on the operator Φ and on the second fundamental form to R guarantee that either Φ ≡ 0 or the integral M tr ΦdM is infinite. We will given some applications. The first one says that if M admits an ¯ integrable distribution whose integrals are minimal submanifolds in M then the volume of M must be infinite. Another application states that ¯ satisfies K ¯ ≤ −c2 , for some c ≥ 0, if the sectional curvature of M m 1 and λ : M → [0, ∞) is a nonnegative C function such that gradient vector of λ and the mean curvature vector H of the immersion satisfy |H R +sp∇λ| ≤ (m − 1)cλ, for some p ≥ 1, then either λ ≡ 0 or the integral λ dM is infinite, for all 1 ≤ s ≤ p. M

1. Introduction ¯ be an isometric immersion of an m-dimensional RieLet f : M m → M ¯ and II the second funmannian manifold M in a Riemannian manifold M damental form of f . Let Φ : T M → T M be a symmetric operator on M of class C 1 . Consider the following definitions: Definition 1.1. The Φ-mean curvature vector field of the immersion f is the normal vector field HΦ : M → T ⊥ M to the immersion f defined by the trace: HΦ = tr {(X, Y ) ∈ T M × T M 7→ II(ΦX, Y )} Note that HΦ coincides with the mean curvature vector if Φ is the identity operator I : T M → T M . Definition 1.2. The divergence of Φ is the tangent vector field on M defined by hdiv Φ, Xi = tr {Y ∈ T M 7→ (∇Y Φ)X = ∇Y (ΦX) − Φ(∇Y X)}, Date: December 21, 2013. 2010 Mathematics Subject Classification. 53C42; 53C40.

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M. BATISTA AND H. MIRANDOLA

for all tangent vector field X : M → T M . Note that if Φ = λI, where λ : M → R is a C 1 function, then div Φ coincides with the gradient vector of λ. It is a well known fact that a complete noncompact minimal submanifold in a Hadamard manifold must have infinite volume (see for instance [9]). Our first theorem says the following: ¯ be an isometric immersion of a complete Theorem 1.1. Let f : M → M ¯ with noncompact manifold M in a complete simply-connected manifold M nonpositive radial curvature with respect to some basis point q0 ∈ f (M ). Let Φ : T M → T M be a positive-semidefinite symmetric operator of class C 1 such that tr Φ(q0 ) > 0. Assume that |HΦ + div Φ| ≤

1 tr Φ, r+ǫ

¯ from q0 . Then the for some ǫ > 0, where r = dM¯ R(· , q0 ) is the distance in M rate of growth of the integral M tr Φ is at least logarithmic with respect the geodesic balls centered at q0 , that is, Z 1 tr Φ > 0, lim inf µ→∞ log(µ) B (q ) µ 0 where BRµ denote the geodesic balls of M centered at q0 . In particular, the integral M tr Φ must be infinite.

Before we enunciate the next results, we will to consider a consequence of Theorem 1.1. Let M be a manifold with nonpositive radial curvature with respect to some base point q0 . It is easy to show that the radial curvature ¯ = M × R with respect to the base point (q0 , 0) of the product manifold M ¯ given by is also nonnegative. Furthermore, the inclusion map j : M → M j(x) = (x, 0) is a totally geodesic embedding. Thus the result below follows directly from Theorem 1.1. Corollary 1.1. Let M be a complete simply-connected manifold with nonpositive radial curvature with respect to some base point q0 . Let Φ : T M → T M be a positive-semidefinite symmetric operator such that tr Φ(q0 ) > 0. Assume that 1 tr Φ, |div Φ| ≤ r+ǫ Rfor some ǫ > 0, where r = dM¯ (· , q0 ). Then the rate of growth of the integral M tr Φ is at least logarithmicRwith respect to the geodesic balls centered at q0 . In particular, the integral M tr Φ is infinite. The next theorem says the following:

INTEGRAL ESTIMATES FOR THE TRACE OF SYMMETRIC OPERATORS

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¯ be an isometric immersion of a complete Theorem 1.2. Let f : M → M ¯ . Let Φ : T M → T M noncompact manifold M in a Hadamard manifold M be a positive-semidefinite symmetric operator of class C 1 . Assume that div Φ = HΦ = 0. Then, for allR q ∈ M , we have that either Φ(q) = 0 or the rate of growth of the integral M tr Φ is at least linear with respect to the geodesic balls of M centered at q, that is, Z 1 tr Φ > 0, lim inf µ→∞ µ B (q) µ where Bµ (q) denotes the geodesic ball of M of radius µ and R centered at q. In particular, either Φ vanishes identically or the integral M tr Φ is infinite.

The following result is a non-direct application of Theorem 1.2. It will be proved in section 3 below. ¯ be an isometric immersion of a complete Corollary 1.2. Let f : M → M ¯ with noncompact manifold M in a complete simply-connected manifold M nonpositive radial curvature with respect to some base point in f (M ). Let D be a k-dimensional integrable distribution on M , with k ≥ 1, such that ¯ . Then the rate of volume each integral of D is a minimal submanifold in M growth of M is at least linear. To state our next applications of Theorems 1.1 and 1.2 we need to recall some notations. Let B : T M → T M be a symmetric operator of class C 1 . We recall that B satisfies the Codazzi equation if the following holds: (∇X B)Y = (∇Y B)X, for all tangent vector fields X and Y on M . The Newton operators Pj = Pj (B), j = 1, . . . , m, associated to B, are the symmetric operators on M defined inductively by: P0 = I; Pj = Sj I − BPj−1 , with j ≥ 1, P where Sj = Sj (B) = i1 0. µ→∞ tr Φ(q0 ) B µ(q ) 0

RMoreover the limit in (1) does not depend of q0 . In particular, the integral M tr Φ is infinite.

The result bellow follows from Theorem 1.5 by considering Φ = λs I, with 1 ≤ s ≤ p, where λ : M → R is a nonnegative C 1 -function. ¯ be an isometric immersion of a complete Corollary 1.3. Let f : M m → M ¯ with sectional curvanon-compact manifold M in a Hadamard manifold M 2 ¯ ≤ −c , for some constant c ≥ 0. Let λ : M → [0, ∞) be a ture satisfying K 1 nonnegative C function satisfying: |λH + p∇λ| ≤ (m − 1)cλ, for some constant p ≥ 1, where H = tr II is the mean curvature vector field of the immersion f . RThen, for all q ∈ M , either λ(q) = 0 or the rate of growth of the integral M λs satisfies Z µ−1 λs > 0, (2) lim inf s µ→∞ λ (q) B (q) µ for all 1 ≤ s ≤ p. Moreover the limit R in (2) does not depend of q. In particular, either λ ≡ 0 or the integral M λs is infinite, for all 1 ≤ s ≤ p.

Let M be a Hadamard manifold with sectional curvature satisfying K ≤ −c2 , for some constant c ≥ 0. It is simple to show that the warped product ¯ = R ×cosh(ct) M is also a Hadamard manifold with sectional curvaspace M ¯ ≤ −c2 . Furthermore the inclusion map i : M → {0}×M ⊂ ture satisfying K ¯ is a totally geodesic embedding. Thus it follows from Corollary 1.3 the M result below: Corollary 1.4. Let M be a Hadamard manifold with sectional curvature satisfying K ≤ −c2 , for some constant c ≥ 0. Let λ : M → R be a nonnegaλ, for some constant p ≥ 1. Then tive C 1 function satisfying |∇λ| ≤ (m−1)c Rp s either λ ≡ 0 or the rate of growth of M λ satisfies Z µ−1 (3) lim inf s λs > 0, µ→∞ λ (q) B (q) µ

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M. BATISTA AND H. MIRANDOLA

for all 1 ≤ s ≤ p. Moreover the limit R in (3) does not depend of q. In particular, either λ ≡ 0 or the integral M λs is infinite, for all 1 ≤ s ≤ p.

Finally we will enunciate our last theorem. We recall that an end E of M is an unbounded connected component of the complement set M − Ω, ¯ has bounded for some compact subset Ω of M . We say that a manifold M geometry if it has sectional curvature bounded from above and injective radius bounded from below by a positive constant. By results of Frensel [9], Cheng, Cheung and Zhou [5] and do Carmo, Wang and Xia [6] we know the following theorem. ¯ be an isometric immersion of a complete Theorem B. Let f : M → M ¯ with bounded geometry. If the non-compact manifold M in a manifold M mean curvature vector of f is bounded in norm then each end of M has infinite volume. Actually, Cheng, Cheung and Zhou [5] improve Theorem B by showing that the volume growth of each end E of M is at least linear, that is vol(Bµ (q) ∩ E) > 0, (4) lim inf µ→∞ µ for all q ∈ E. Moreover the limit (4) does not depend of of q (see Proposition 2.1 of [5]). Our last theorem says the following: ¯ be an isometric immersion of a complete Theorem 1.6. Let f : M m → M ¯ with bounded geometry. Let E non-compact manifold M in a manifold M be an end of M and λ : E → R a nonnegative C 1 function. Assume that the mean curvature vector field H = tr II of the immersion f satisfies |Hλ + p∇λ| ≤ κ λ in E, for some constant κ ≥ 0. Then it holds that lim x→∞ λ(x) = 0 or the integrals x∈E R s λ are infinite, for all 1 ≤ s ≤ p. E

¯ = Note that if M has bounded geometry then the product manifold M M × R also has bounded geometry. Since the inclusion map i : M → ¯ is a totally geodesic embedding, the result below follows from M × {0} ⊂ M Theorem 1.6.

Corollary 1.5. Let E be an end of a complete non-compact manifold with bounded geometry. Let λ : E → [0, ∞) be a nonnegative C 1 function. Assume that |∇λ| ≤ κλ in E, for some constant κ ≥ 0. Then it holds that lim x→∞ λ(x) = 0 or the integrals x∈E R p λ are infinite, for all p ≥ 1. E

INTEGRAL ESTIMATES FOR THE TRACE OF SYMMETRIC OPERATORS

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2. Preliminaries ¯ be an isometric immersion of an m-dimensional RieLet f : M m → M ¯ . For the sake of simplicmannian manifold M in a Riemannian manifold M ity, henceforth we shall make the usual identification of the points f (q) with q and the vectors dfq v with v, for all q ∈ M and v ∈ Tq M . Let Φ : T M → T M be a symmetric operator of class C 1 . We consider the following definition: ¯ of class Definition 2.1. The Φ-divergence of a vector field X : M → T M 1 C is given by the following: n
o ¯ ZX T , DΦ X = tr Z ∈ T M 7→ Φ ∇

Note that if Φ = I : T M → T M is the identity operator then DΦ X coincides with the divergence divM X.

Let u = (u1 , . . . , um ) be a local coordinate system on M . Let { ∂u∂ 1 , . . . , ∂u∂m } and {du1 , . . . , dum } be the frame and coframe associated to u, respectively. Using the Einstein’s summation convention, let h , i = gij dui ⊗ duj be the metric of M . The Cheng-Yau square operator [4] associated to the symmet

∂ ∂ i j ric (0, 2)-tensor φ = φij du ⊗ du , where φij = Φ( ∂ui ), ∂uj = Φki gkj , is defined by φ f = DΦ (∇f ).

(5)

It was proved in [4] that the operator φ is self-adjoint on the space of the all Sobolev functions with null trace if, and only if, the covariant derivative ¯ be a vector field of class of Φ satisfies Φij,i = 0, for all j. Let X : M → T M C 1 and write X T = X j ∂u∂ j . Using that ∇ ∂ X T = ∇ ∂ (X j ∂u∂ j ) = X,ij ∂u∂ j , ∂ui

we have that

∂ui

∂ ∂ ) = X,ij Φkj k . j ∂u ∂u j i ∂ T T Thus we have that DΦ (X ) = X,i Φj . Since Φ(X ) = X j Φ( ∂u∂ j ) = X j Φij ∂u i, we obtain Φ(∇

∂ ∂ui

X T ) = Φ(X,ij

div M (Φ(X T )) = (X j Φij ),i = X,ij Φij + X j Φij,i

(6)

= DΦ X T + (div Φ)∗ (X T ), where (div Φ)∗ the 1-form defined by (div Φ)∗ = Φij,i duj . Using that (∇ Φkj,i ∂u∂ k we have that (∇ plies that (7)

∂ ∂ui

Φ)X T = X j (∇

∂ ∂ui

∂ ∂ui

∂ Φ) ∂u j =

∂ j k ∂ Φ) ∂u j = X Φj,i ∂uk . This im-

 (div Φ)∗ (X T ) = X j Φij,i = tr Y ∈ T M 7→ (∇Y Φ)X T

 = div Φ, X T .

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M. BATISTA AND H. MIRANDOLA

¯ be a C 1 vector field and f : M → R Proposition 2.1. Let X : M m → T M 1 a C function. Then the following statements hold: (A) DΦ X = DΦ X T − hHΦ , Xi;  (B) DΦ (f X) = f DΦ X + Φ(X T ), ∇f ; (C) DΦ X = div M (Φ(X T )) − h(HΦ + div M Φ), Xi . Proof. Write X = X T + X N , where X N is the orthogonal projection of X to T ⊥ M . Let {e1 , . . . , em } be a local orthonormal frame of M . We have that m m m X

 X

 X

 T ¯ e X)T , ei = ¯ e X N )T , ei DΦ X = Φ(∇ Φ(∇ X ), e + Φ(∇ ei i i i i=1

i=1

= DΦ X T −

m X

II(ei , Φei ), X

i=1

i=1

 N

= DΦ X T − hHΦ , Xi ,

which proves Item (A). Now, m m X 

 X

¯ e X)T , ei ¯ e f X)T , ei = ei (f )Φ(X T ) + f Φ(∇ Φ(∇ DΦ (f X) = i i i=1

i=1

= f DΦ X +

m X i=1



 ei (f ) Φ(X T ), ei = f DΦ X + Φ(X T ), ∇f ,

which proves Item (B). Using (A), (6) and (7) we obtain that DΦ X = div M (ΦX T ) − h(HΦ + div M Φ), Xi ,

(8)

which proves Item (C).



Remark 1. By (C) and the divergence theorem we have that Z Z Z

 T h(HΦ + div M Φ), Xi , ΦX , ν − DΦ X = (9) D

∂D

D

where D is a bounded domain with Lipschitz boundary ∂D and ν is the exterior conormal along ∂D. Equation (9) holds in the sense of the trace. Example 2.1. Take p ∈ M and let x = (x1 , . . . , xn ) be a coordinate system ¯ . Write ∂ Ti = ( ∂ i )T = aj ∂ j and in a normal neighborhood V of p in M i ∂x ∂x ∂x j ∂ ∂T ¯ ) = φ . We have that Φ( ∂x i i ∂xj   ∂T ∂ T k ¯l ¯ ¯ l φ¯s ∂ , Φ ∇ ∂T = a ) = aki Γ Γ Φ( i kj kj l j l ∂x ∂xs ∂xi ∂x ¯ k are the Christoffel symbols of the Riemannian connection ∇ ¯ of M ¯. where Γ ij ∂ k l i ¯ φ¯ . Using (B), it holds the following: This implies that DΦ ( j ) = a Γ ∂x

DΦ (h

i

kj l

∂ ¯ l φ¯i + φ¯kj ∂h , ) = haki Γ kj l j ∂x ∂xk

INTEGRAL ESTIMATES FOR THE TRACE OF SYMMETRIC OPERATORS

9

¯ for all h ∈ C 1 (M ). As a particular case, consider the vector field Y = r ∇r, ¯ → [0, ∞) is the distance function r(q) = d ¯ (q, q0 ), for some where r : M M ¯ . Using that r 2 (q) = (x1 (q))2 + . . . + (xn (q))2 , for all q ∈ V , we have q0 ∈ M ¯ 2 = xj (q) ∂ j . Using that Y = xj ∂ j and tr M Φ = φ¯i , we that Y (q) = 21 ∇r i ∂x ∂x obtain that ∂xj ¯ l φ¯i . = tr M Φ + xj aki Γ kj l ∂xk ¯ is flat then DΦ (Y ) = tr Φ. Since Y = 2−1 ∇r ¯ 2 , using (A) In particular, if M and (5), we obtain that

 ¯ 2−1 Φ r 2 = DΦ Y + r HΦ , ∇r

 ¯ + x j ak Γ ¯ l φ¯i . = tr Φ + r HΦ , ∇r ¯ l φ¯i + φ¯kj DΦ (Y ) = xj aki Γ kj l

i

kj l

Let K : R → R be an even continuous function. Let h be a solution of the Cauchy problem  ′′ h + Kh = 0, (10) h(0) = 0, h′ (0) = 1.

Let I = (0, r0 ) be the maximal interval where h is positive. ¯ be a Riemannian manifold and B = Br (q0 ) the geodesic ball of M ¯ Let M 0 with center q0 and radius r0 > 0. Consider the radial vector field ¯ X = h(r)∇r, defined on B ∩ V , where r = dM¯ (· , q0 ), for some fixed point q0 ∈ M , and V ¯ . The result below generalizes Example is a normal neighborhood of q0 in M 2.1 and will be useful is the proof of Theorem 1.1. ¯ be an isometric immersion of a manProposition 2.2. Let f : M m → M ¯ . Assume that the radial curvature of M ¯ with ifold M in the manifold M ¯ respect to the basis point q0 ∈ M satisfies Krad ≤ K(r) in B ∩ V . Let Φ : T M → T M be a symmetric operator. Assume that one of the following conditions hold: (i) Φ is positive-semidefinite; or ¯ rad = K(r), (ii) K Then, it holds that (11)

DΦ X ≥ h′ (r)tr Φ.

Moreover, the equality occur if (ii) holds. Proof. Take q ∈ M ∩ V . Let ξ = {e1 , . . . , em } be an orthonormal basis of Tq M by eigenvectors of Φ and {λ1 , . . . , λm } the corresponding eigenvalues.

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M. BATISTA AND H. MIRANDOLA

We extend the basis ξ to an orthonormal frame on a neighborhood of q in M . Then, at the point q, we have ¯ = DΦ ∇r

m X

i=1

m m  X

 X ¯ ei = ¯ T , ei = λi (HessM¯ r)q (ei , ei ), λi ∇ei ∇r, Φ(∇ei ∇r) i=1

i=1

where HessM¯ r is the Hessian of r. ˜ be the Euclidean ball of Rm of radius r0 and center at the origin Let M 0 endowed with the metric g˜, which in polar coordinates can be written as g˜ = dρ2 + h(ρ)2 dω 2 , where dω 2 represents the standard metric on the Euclidean unit sphere S m−1 . Consider the distance function r˜ = dM˜ (· , 0). Then, for x = ρω with ρ > 0 and ω ∈ S m−1 , the Hessian of r˜ satisfies: Hess(˜ r) =

h′ (˜ g − d˜ r ⊗ d˜ r ), h

′′ r ) ˜ Furthermore, the function K(˜ r ) = − hh(˜(˜ r ) is the radial curvature of M with ¯ with respect to the basis point 0. Thus since the radial curvature of M ¯ respect to some basis point q0 satisfies Krad ≤ K(r) it follows from the Hessian comparison theorem (see Theorem A page 19 of [10]) the following:  h′ (r)  1 − h∇r, ei i2 . (12) (HessM¯ r)(ei , ei ) ≥ h(r) ¯ rad = K(r). Since hΦei , ej i = Moreover, the equality in (12) holds when K

q

λi δij , for all i, j, we obtain (13)

¯ = h(r)DΦ ∇r ¯ + h′ (r) h∇r, Φ∇ri DΦ X = DΦ (h(r)∇r) m X λi (HessM¯ r)q (ei , ei ) + h′ (r) h∇r, Φ∇ri . = h(r) i=1

Using that the hypothesis (i) and (ii), it follows from (12) and (13) that m  h′ (r) X  (14) DΦ X ≥ h(r) λj 1 − h∇r, ej i2 + h′ (r) h∇r, Φ∇ri h(r) j=1

′

= h (r) (tr Φ − h∇r, Φ∇ri) + h′ (r) h∇r, Φ∇ri = h′ (r)tr Φ. ¯ rad = K(r). Moreover, the equality in (14) holds when K Corollary 2.1. Under the hypotheses of Proposition 2.2 we have that Z Z
h′ (r)tr Φ − h(r)|HΦ + div M Φ| , h(r) hΦ∇r, νi ≥ ∂D

D



INTEGRAL ESTIMATES FOR THE TRACE OF SYMMETRIC OPERATORS

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where D is a bounded domain compactly contained in V ∩ B with Lipschitz boundary ∂D, and ν is the exterior conormal along ∂D. Proof. Using (C) we have that div M (ΦX T ) = DΦ X+hHΦ + div Φ, Xi. Since ¯ = 1, using (11) and the divergence theorem (see for instance [7]), we |∇r| obtain that Z Z Z

 ′ ¯ (15) h(r) HΦ + div Φ, ∇r h (r)tr Φ + h(r) hΦ∇r, νi ≥ D ∂D ZD
h′ (r)tr Φ − h(r)|HΦ + div M Φ| . ≥ D

Corollary 2.1 is proved.



¯ q the injectivity radius of M ¯ at the point q0 and Bµ (q) We denote by R 0 the geodesic ball of M with center q and radius µ. Let α : [0, ∞) → [0, ∞) be a nonnegative C 1 function. We consider the following positive number: o n h′ (t)2 h′ (t) > α(t) and α′ (t) ≥ − − K(t) . (16) µK,α = sup t ∈ (0, r0 ); h(t) h(t)2 3. proof of Theorem 1.1, Theorem 1.2 and Corollary 1.2. The main tool of this section is the following result: ¯ be an isometric immersion of a complete Theorem 3.1. Let f : M → M ¯ . Assume that the radial curvature noncompact manifold M in a manifold M ¯ ¯ rad ≤ K(r), where r = of M with base point in some q0 ∈ f (M ) satisfies K dM¯ (· , q0 ) and K : R → R is an even continuous function. Let Φ : T M → T M be a positive-semidefinite symmetric operator such that tr Φ(q0 ) > 0. Assume further that (17)

|HΦ + div Φ| ≤ α(r) tr Φ,

where α : [0, ∞) → [0, ∞) is a nonnegative C 1 function. Then, for each ¯ q }, there exists a positive constant Λ = Λ(q0 , µ0 , M ) 0 < µ0 < min{µK,α , R 0 satisfying Z Z µ

tr Φ ≥ Λ

Bµ (q0 )

−

e

Rτ

µ0

α(s)ds

dτ,

µ0

¯ q }. for all µ0 ≤ µ < min{µK,α , R 0 ¯ 0 = min{µK,α , R ¯ q } and let Bµ = Bµ (q0 ). Note that Proof. Take 0 < µ < R 0 the distance function ρ = dM (· , q0 ) satisfies r ≤ ρ. This implies that Bµ is ¯ with center q0 and radius R ¯0. contained in the geodesic ball BR¯ 0 (q0 ) of M Since M is a complete noncompact manifold and ρ is a Lipschitz function we obtain that the ball Bµ is a bounded domain of M with Lipchitz boundary ∂Bµ 6= ∅. Since |∇ρ| = 1 a.e. in Bµ , using Corollary 2.1, equation (17) and

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M. BATISTA AND H. MIRANDOLA

the coarea formula (see for instance [7] or Theorem 3.1 of [8]), we obtain that Z  ′ Z h (r) − α(r)h(r)  (18) h(r)tr Φ h(r) hΦ∇r, νi ≥ h(r) Bµ ∂Bµ Z µZ   h′ (r) − α(r) h(r)tr Φ, = h(r) 0 ∂Bτ ¯ 0 , where ν is the exterior conormal along for almost everywhere 0 < µ < R ∂Bµ . Take q ∈ M and let {e1 , . . . , em } ⊂ Tq M be an orthonormal basis by eigenvectors of Φ at the point q. Consider {λ1 , . . . , λm } the corresponding eigenvalues. Since Φ is positive-semidefinite we have that λi ≥ 0, for all i. Since |∇r| ≤ 1, using Cauchy-Schwartz inequality, we obtain (19) hΦ∇r, νi =

m X

λi h∇r, ei i hν, ei i ≤

m X

λi |∇r||ν| = (tr Φ)|∇r| ≤ tr Φ.

i=1

i=1

Thus, it follows from (18) that Z Z h(r)tr Φ ≥ (20)

0

∂Bµ

µZ

∂Bτ

 h′ (r) h(r)

 − α(r) h(r)tr Φ,

¯0 . for a.e. 0 < µ < R We define the following functions R ¯0 ) 7→ F (µ) = Φ; F : µ ∈ (0, R ∂B h(r)tr   ′ R µ Rµ (21) h (r) ¯0 ) 7→ G(µ) = − α(r) h(r)tr Φ, G : µ ∈ (0, R 0 ∂Bτ h(r)

It follows from (20) that

F (µ) ≥ G(µ),

(22) ¯0 . for a.e. 0 < µ < R ′

2

(t) Note that α′ (t) ≥ − hh(t) 2 −K(t) is equivalent to say that h′ (t)



′ h′ (t) −α(t) h(t)

≤ 0.

Thus, by hypothesis, the function t ∈ (0, µK,α ) 7→ h(t) − α(t) is positive and non-increasing. Since the function r = dM¯ ( · , q0 ) satisfies r ≤ τ in ∂Bτ we ′ (τ ) ′ (r) ¯ − α(r) ≥ hh(τ have that hh(r) ) − α(τ ) > 0 in ∂Bτ , for all 0 < τ ≤ R0 . This ¯ 0 , since G ≥ 0, G is nondecreasing implies that G(µ) > 0, for all 0 < µ < R and G(µ) > 0, for all µ > 0 sufficiently small (recall that tr Φ(q0 ) > 0). Using (21) and (22), we have that  ′   ′  h (µ) h (µ) ′ G (µ) = − α(µ) F (µ) ≥ − α(µ) G(µ), h(µ) h(µ)

INTEGRAL ESTIMATES FOR THE TRACE OF SYMMETRIC OPERATORS

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¯ 0 . Thus we obtain for a.e. 0 < µ < R  d  G′ (µ) h′ (µ) d ln G(µ) = ≥ − α(µ) = ln h(µ) − α(µ), dµ G(µ) h(µ) dµ ¯ 0 . Integrating (23) over (µ0 , µ), with 0 < µ0 < µ, we for a.e. 0 < µ < R obtain that Z µ G(µ) h(µ) ln ≥ ln − α(s)ds. G(µ0 ) h(µ0 ) µ0

(23)

This implies that (24)

−

G(µ) ≥ Λ(q0 , µ0 , M )h(µ)e

Rµ

µ0

α(s)ds

,

0) where Λ = Λ(µ0 , q0 , M ) = G(µ h(µ0 ) . R ¯ 0 . Since Now, we define the function f (µ) = Bµ tr Φ, with 0 < µ < R h(r) ≤ h(µ) in ∂Bµ it follows from (22), (24) and the coarea formula that Z Z R F (µ) G(µ) 1 − µ α(s)ds ′ h(r)tr Φ = ≥ ≥ Λ e µ0 , tr Φ ≥ f (µ) = h(µ) ∂Bµ h(µ) h(µ) ∂Bµ

¯ 0 . Since f (µ0 ) ≥ 0 we have that for a.e. 0 < µ < R Z µ Rτ Z α(s)ds − dτ. tr Φ = f (µ) ≥ Λ e µ0 Bµ

µ0

This concludes the proof of Theorem 3.1.



Now we are able to prove Theorem 1.1, Theorem 1.2, Corollary 1.2 and Theorem 1.3. 3.1. Proof of Theorem 1.1. First we observe that the injectivity radius ¯ q = +∞, since M ¯ has nonpositive radial curvature with base point q0 . R 0 1 We take the functions K(t) = 0, with t ∈ R, and α(t) = t+ǫ , with t ≥ 0. The function h(t) = t, with t > 0, is the maximal positive solution of ¯ rad ≤ 0 = K(r) and (10). Furthemore, we have that µK,α = ∞. Since K tr Φ(q0 ) > 0, Theorem 3.1 applies. Thus it holds that Z µ Rτ Z  µ+ǫ  ds − tr Φ ≥ Λ e µ0 s+ǫ dµ = Λ(µ0 + ǫ) log , µ0 + ǫ µ0 Bµ for all 0 < µ0 < µ, where Λ is a positive constant depending only on q0 , µ0 and M . This implies that Z 1 tr Φ > 0. lim inf µ→∞ log(µ) B (q ) µ 0 Theorem 1.1 is proved.

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M. BATISTA AND H. MIRANDOLA

3.2. Proof of Theorem 1.2. Similarly as in the proof of Theorem 1.1 we ¯ q . Consider the function K(t) = 0, with t ∈ R, and α(t) = 0, have that R 0 with t ≥ 0. RWe have that µK,α = +∞ and Theorem 3.1 applies. Thus we obtain that Bµ (q0 ) tr Φ ≥ Λ(µ − µ0 ), for all 0 < µ0 < µ, where Λ is a positive constant depending only on q0 , µ0 and M . This implies that Z 1 tr Φ ≥ Λ > 0. lim inf µ→∞ µ B (q ) µ 0 Theorem 1.2 is proved. ¯1 , . . . , E¯k } 3.3. Proof of Corollary 1.2. Fix q ∈ M . Let {E1 , . . . , Em } and {E be orthonormal frames of T M and

D defined in a neighborhood U of q in ¯l , for all v ∈ T U , we obtain that M , respectively. Since PD (v) = v, E¯l E div PD =

=

m X

(∇Ei PD )Ei i=1 k  m X X Ei

i=1 l=1

=

=

k m X X

i=1 l=1 k X



=

m X

¯l Ei , E

=

 



¯l ¯l + Ei , E ¯ l ∇E E ¯l − ∇E Ei , E ¯l E E i i

k X

¯l ∇Pm hEi ,E¯l iEi E i=1

k X

¯l . ∇E¯l E

l=1

l=1

(25)




 

¯l ¯l + Ei , E ¯ l ∇E E ¯l E Ei , ∇Ei E i

¯l + (div M (E¯l ))E

k X

∇Ei (PD (Ei )) − PD (∇Ei Ei )

i=1

¯l + (div M (E¯l ))E

l=1

l=1

Since the distribution D is integrable, there exists an embedded submani˜k } fold S ⊂ M satisfying q ∈ S and Tx S = D(x), for all x ∈ S. Let {E˜1 , . . . , E be an orthonormal frame, defined in a small neighborhood U of q in S, that is geodesic at q with respect to the connection of S, namely, (26)

˜s )q = PD (∇ ˜ E ˜ (∇SE˜ E El s )q = 0, l

˜k+1 , . . . , E ˜m } be an orthonormal frame for all l, s = 1, . . . , k. Now, let {E ⊥ of the normal bundle T S defined in a small neighborhood of q in S, that ˜1 , . . . , E ˜m } to an we can also assume to be U . We extend the frame {E orthonormal frame defined in a small tubular neighborhood W of U in M by parallel transport along minimal geodesics from U to the points of W . ˜l )x = 0, for all x ∈ U , l = 1, . . . , k and In particular, it holds that (∇E˜β E

INTEGRAL ESTIMATES FOR THE TRACE OF SYMMETRIC OPERATORS

15

β = k + 1, . . . , m. This fact, togheter with (26), imply that (27) (div M (E˜l ))q =

k D X i=1

m D E E X ˜l )q , E˜β (q) = 0, ˜l )q , E˜i (q) + (∇E˜β E (∇SE˜ E i

β=k+1

for all l = 1, . . . , k. Thus, by (25), (26) and (27) we obtain that (28) k k m D E X X X S ˜ ˜l )q , E˜β (q) E ˜β (q) = (∇E˜l E (div PD )q = IIM (El (q), E˜l (q)), l=1 β=k+1

l=1

S is the second fundamental form of the submanifold S in M . where IIM ˜ of the restriction On the other hand, the second fundamental form II ¯ f |S : S → M is given by: S S ˜ (v, v) + II(v, v) = IIM (v, v) + II(PD v, v), (29) II(v, v) = IIM

for all v ∈ Tx S = D(x), with x ∈ S, where II denotes the second fundamental ¯ . Thus, by (28) and (29), we have that form of the immersion f : M → M ˜ = div PD + HP . (30) tr II ¯ is minimal. Thus, by By hypothesis the isometric immersion f |S : S → M ˜ (30), it holds that tr II = div PD + HP = 0. Since tr PD = k ≥ 1 itR follows from Theorem 1.2 that the rate of growth of the volume vol(M ) = k1 M tr PD is at least linear with respect to the geodesic balls centered to any point of M . Corollary 1.2 is proved. 4. Proof of Theorem 1.3 and Theorem 1.4 Before we prove Theorem 1.3 we need some preliminaries. Let W m be an m-dimensional vector space and T : W → W a symmetric linear operator on W . Consider the Newton operators Pj (T ) : W → W , j = 0, . . . , m, associated to T . It is easy to shows that each Pj (T ) is a symmetric linear operator with the same eigenvectors of T . Let {e1 , . . . , em } be an orthonormal basis of W by eigenvectors of T and {λ1 , . . . , λm } the corresponding eigenvalues. Let Wj = {ej }⊥ , j = 1, . . . , m, be the orthogonal hyperplane to ej and consider Tj = T |Wj : Wj → Wj . The two lemmas below were proved for the case that T is the shape operator A(p) associated to a hypersurface of a Riemannian manifold evaluated at some point p (see Lemma 2.1 of [2] and Proposition 2.4 of [1], respectively). The proof in the general case follows exactly the same steps. Lemma 4.1. For each 1 ≤ j ≤ m − 1, the following items hold: (a) Pj (T )ek = SjP (Tk )ek , for each 1 ≤ k ≤ m; (b) tr (Pj (T )) = m k=1 Sj (Tk ) = (m − j)Sj (T );

16

M. BATISTA AND H. MIRANDOLA

P (c) tr (T Pj (T )) = Pm k=1 λk Sj (Tk ) = (j + 1)Sj+1 (T ); 2 (d) tr (T 2 Pj (T )) = m k=1 λk Sj (Tk ) = S1 (T )Sj+1 (T ) − (j + 2)Sj+2 (T ).

Lemma 4.2. Assume that Sj+1 (T ) = 0, for some 1 ≤ j ≤ n − 1. Then Pj (T ) is semidefinite. We also need of the following lemmas: Lemma 4.3. Assume that Sj−1 (T ) = Sj (T ) = 0, for some 2 ≤ j ≤ m. Then the rank of T satisfies rk(T ) ≤ j − 2. Proof. If T = 0 then there is nothing to prove since rk(T ) = 0 ≤ m − 2. Thus we can assume that T 6= 0. We will prove Lemma 4.3 by induction on m = dim W . First we assume that m = 2. Since kT k2 := λ21 + λ22 = (λ1 + λ2 )2 − 2λ1 λ2 = S1 (T )2 − 2S2 (T ) = 0 it follows that T = 0. Now we assume that Lemma 4.3 is true for any symmetric operator Q : V k → V k defined on a k-dimensional vector space V , with 2 ≤ k ≤ m − 1. Since Sj−1 (T ) = Sj (T ) = 0, for some 2 ≤ j ≤ m, it follows from Lemma 4.2 that the operators Pj−2 (T ) and Pj−1 (T ) are semidefinite. Thus using that tr (Pj−1 (T )) = (m − j + 1)Sj−1 (T ) = 0 it follows that Pj−1 (T ) = 0. Furthermore, the operator T 2 Pj−2 (T ) is also semidefinite with trace satisfying tr (T 2 Pj−2 (T )) = S1 (T )Sj−1 (T ) − jSj (T ) = 0, which implies that T 2 Pj−2 = 0. Since (T 2 Pj−2 )ek = λ2k Sj−2 (Tk )ek = 0 we obtain that λk = 0 or Sj−2 (Tk ) = 0. Thus, using that Sj−1 (Tk ) = hPj−1 (T )ek , ek i = 0 and dim(Wk ) = m − 1, we obtain by the induction assumption that λk = 0 or rk(Tk ) ≤ j − 3. Since T 6= 0 there exists some eigenvalue λk 6= 0. Thus we obtain that rk(Tk ) ≤ j − 3 which implies that rk(T ) ≤ j − 2.  Lemma 4.4. Let B : T M → T M be a symmetric operator of class C 1 that satisfies the Codazzi equation. Then it holds that div (Pj (B)) = 0. Proof. We denote by Pj = Pj (B), with j = 1, . . . , m. Take p ∈ M and let {E1 , . . . , Em } be an orthonormal frame defined on an neighborhood V of p

INTEGRAL ESTIMATES FOR THE TRACE OF SYMMETRIC OPERATORS

17

in M , geodesic at p. We have that m m X X (∇Ei Sj I − BPj−1 )Ei (∇Ei Pj )Ei = div Pj = i=1

i=1

=

m X

Ei (Sj )Ei − ∇Ei (BPj−1 )Ei

i=1

= ∇(Sj ) −

m X




(∇Ei B)Pj−1 (Ei ) + B(∇Ei Pj−1 )Ei

i=1

= ∇(Sj ) − B(div Pj−1 ) −

(31)




m X (∇Ei B)Pj−1 (Ei ). i=1

C1

Let X be a vector field on M . Since (∇X B) is a symmetric operator and (∇Ei B)X = (∇X B)Ei , for all i = 1, . . . , m, we obtain that (32)

m X

h(∇Ei B)Pj−1 (Ei ), Xi =

m X

hPj−1 (Ei ), (∇X B)Ei i

i=1

i=1


= tr Pj−1 (∇X B) .


It was proved by Reilly [11] (see Lemme A of [11]) that tr Pj−1 (∇X B) = h∇(Sj ), Xi. Thus, using (32), we obtain that (33)

m X

(∇Ei B)Pj−1 (Ei ) = ∇(Sj ).

i=1

Using (31) and (33), we obtain that (div Pj )p = (∇Sj )(p) − B(div Pj−1 )p − (∇Sj )(p) = −B(div Pj−1 )p . Since P0 = I we obtain by recurrence that div Pj = (−1)j B j (div I) = 0. This concludes the proof of Lemma 4.4.  Now, we are able to prove Theorem 1.3 and Theorem 1.4. 4.1. Proof of Theorem 1.3. Since Sj+1 (B) = 0 it follows from Lemma 4.2 that the operator Pj (B(p)) : Tp M → Tp M is semidefinite at each point p ∈ M . Since Sj does not change of sign we obtain that Φ = ǫPj is positivesemidefinite, for some constant ǫ ∈ {−1, 1}. Since B satisfies the Codazzi equation it follows from Lemma 4.4 that div Φ = ǫ div Pj = 0. Since |HΦ + 1 ¯ from q0 and , where r is the distance function of M div Φ| = |HPj | ≤ r+ǫ tr Φ(q0 ) = |tr Pj (q0 )| = (m − j)|Sj (B(q Theorem 1.1 to R 0 ))| > 0 we can apply R conclude that the rate of growth of M tr Φ = (m − j) M |Sj (B)| is at least logarithmic with respect to the geodesic balls of M centered at q0 . Theorem 1.3 is proved.

18

M. BATISTA AND H. MIRANDOLA

4.2. Proof of Theorem 1.4. Since Sj+1 = Sj+1 (A) = 0 it follows from Lemma 4.2 that the operator Pj (A(p)) : Tp M → Tp M is semidefinite at each point p ∈ M . Since Sj does not change of sign we obtain that Φ = ǫPj is positive-semidefinite, for some constant ǫ ∈ {−1, 1}. Since the shape operator A satisfies the Codazzi equation it follows from Lemma 4.4 that div Φ = ǫ div Pj = 0. Since |HΦ + div Φ| = |HPj | = (j + 1)Sj+1 = 0 and tr Φ(q0 ) = |tr Pj (q0 )| = (m − j)|Sj (A(q0 ))| > 0 we 1.2 to R can apply Theorem R conclude that the rate of growth of the integral M tr Φ = (m − j) M |Sj (A)| is at least linear with respect to the geodesic balls of M centered at q0 . Theorem 1.4 is proved. 5. Proof of Theorem 1.5, Corollary 1.3 and Theorem 1.6 The main tool of this section is the following result: ¯ be an isometric immersion of a complete Theorem 5.1. Let f : M → M ¯ . Assume that the radial curvature noncompact manifold M in a manifold M ¯ ¯ rad ≤ K(r), where r = of M with base point in some q0 ∈ f (M ) satisfies K dM¯ (· , q0 ) and K : R → R is an even continuous function. Let Φ : T M → T M be a positive-semidefinite symmetric operator such that tr Φ(q0 ) > 0. Assume further that (34)

|HΦ + div Φ| ≤ α(r) tr Φ

and

m|Φ∇r| ≤ tr Φ,

C 1 -function.

where α : [0, ∞) → [0, ∞) is a nonnegative Then Z µ Z Rτ h(τ )m−1 e−m 0 α(s)ds dτ. tr Φ ≥ m tr Φ(q0 ) Bµ (q0 )

0

¯ q }, where h : (0, r0 ) → (0, ∞) is the maximal for all 0 < µ < min{µK,α , R 0 positive solution of (10). Proof. By following exactly the same steps as in the proof of Theorem 3.1 we obtain that Z µZ Z   h′ (r) − α(r) h(r)tr Φ, h(r) hΦ∇r, νi ≥ (35) h(r) 0 ∂Bτ ∂Bµ ¯ 0 = min{µK,α , R(q ¯ 0 )}, where Bµ = Bµ (q0 ) for almost everywhere 0 < µ < R and ν is the exterior conormal along ∂D. Since |ν| = 1 and |Φ∇r| ≤ trmΦ , using Cauchy-Schwartz inequality, we obtain that hΦ∇r, νi ≤ trmΦ . Using (35) we obtain Z µZ Z  h′ (r)  h(r)tr Φ ≥ m (36) − α(r) h(r)tr Φ, h(r) 0 ∂Bτ ∂Bµ ¯0 . for a.e. 0 < µ < R

INTEGRAL ESTIMATES FOR THE TRACE OF SYMMETRIC OPERATORS

19

Consider the following functions R ¯0 ) 7→ F (µ) = Φ F : µ ∈ (0, R ∂B h(r)tr   ′ R µ Rµ h (r) ¯0 ) 7→ G(µ) = − α(r) h(r)tr Φ. G : µ ∈ (0, R h(r) 0 ∂Bτ

It follows by (36) that

F (µ) ≥ m G(µ), ¯ for a.e. µ ∈ (0, R0 ). ¯ 0 , since G ≥ 0, G is nondecreasing Note that G(µ) > 0, for all 0 < µ < R and G(µ) > 0, for µ > 0 sufficiently small (recall that tr Φ(q0 ) > 0). Thus, by (37), we obtain   h′ (µ)   h′ (µ) − α(µ) F (µ) ≥ m − α(µ) G(µ), G′ (µ) = h(µ) h(µ) ¯ for a.e. 0 < µ < R0 . This implies that  h′ (µ)   d   d G′ (µ) ln G(µ) = ≥m − α(µ) = m ln h(µ) − α(µ) dµ G(µ) h(µ) dµ   d (38) = ln h(µ)m − mα(µ), dµ ¯0 ). Integrating (38) over (µ0 , µ), with 0 < µ0 < µ, we for a.e. µ ∈ (0, R obtain that Z µ  G(µ)   h(µ)m  ln ≥ ln −m α(s)ds. G(µ0 ) h(µ0 )m µ0 Thus, we obtain Rµ G(µ0 ) m −m µ0 α(s)ds , h(µ) e (39) G(µ) ≥ h(µ0 )m ¯0 . for all 0 < µ0 < µ < R ′ ¯0 ) 7→ h (µ) − α(µ) is Using that r ≤ µ in Bµ and the function µ ∈ (0, R (37)

h(µ)

non-decreasing we have from the coarea formula that Z  ′   h′ (µ) Z h (r) tr Φ, − α(r) h(r)tr Φ ≥ − α(µ) G(µ) = h(r) h(µ) Bµ Bµ ¯ 0 . Since limt→0 h(t) = h′ (0) = 1 and h(0) = 0 we obtain for all 0 < µ < R t  1 Z    m
µ0 G(µ0 ) ′ lim h (r) − α(r)h(r) tr Φ ≥ lim lim µ0 →0 µm µ0 →0 h(µ0 ) µ0 →0 h(µ0 )m Bµ0 0
′ (40) = h (0) − α(0)h(0) tr Φ(q0 ) = tr Φ(q0 ). Thus, using (39), (40) and taking µ0 → 0, we obtain that (41)

G(µ) ≥ tr Φ(q0 )h(µ)m e−m

Rµ 0

α(s)ds

,

20

M. BATISTA AND H. MIRANDOLA

¯0. for all 0 < µ < R Now we consider the function ¯0 ) 7→ f (µ) = µ ∈ [0, R

Z

tr Φ. Bµ

Since h(r) ≤ h(µ) in ∂Bµ and F (µ) ≥ m G(µ), using the coarea formula and (41), we obtain Z Z F (µ) G(µ) 1 h(r)tr Φ = ≥m tr Φ ≥ f ′ (µ) = h(µ) ∂Bµ h(µ) h(µ) ∂Bµ ≥ m tr Φ(q0 )h(µ)m−1 e−m

Rµ 0

α(s)ds

.

Since f (0) = 0, by integration f ′ (µ) on (0, µ), we have that Z µ Z Rτ tr Φ = f (µ) ≥ m tr Φ(q0 ) h(τ )m−1 e−m 0 α(s)ds dµ. Bµ

0

This concludes the proof of Theorem 5.1.



Now we are able to prove Theorem 1.5 and Theorem 1.6. 5.1. Proof of Theorem 1.5. First we observe that the injectivity radius ¯ at the point q0 satisfies R ¯ q = +∞ since the radial curvature of M ¯ of M 0 with base point q0 is nonpositive. We consider constant functions K(t) = −c2 and α(t) =

(m − 1)c , m

for all t. The maximal positive solution of (10) is given by h(t) = with t > 0. Since cosh(t) ≥ sinh(t), for all t ≥ 0, we obtain h′ (t) = cosh(c t) > (m−1)c m h(t) and
′ (t)2 ′ 2 0 = α (t) ≥ −c (coth(c t))2 − 1 = − hh(t) 2 − K(t),

1 c

sinh(c t),

for all t > 0, which implies that µK,α = ∞. Thus, using Theorem 5.1, we obtain that Z µ Z m sinh(c τ )m−1 e−(m−1)c τ dτ tr Φ ≥ m−1 tr Φ(q0 ) c 0 Bµ Z µ m = (1 − e−2cτ )m−1 dτ tr Φ(q0 ) (2 c)m−1 Z0 µ m ≥ (42) (1 − (m − 1)e−2cτ )dτ. tr Φ(q0 ) (2 c)m−1 0

INTEGRAL ESTIMATES FOR THE TRACE OF SYMMETRIC OPERATORS

21

The last inequality follows from the Bernoulli’s inequality since e−2cτ < 1. This implies that Z m µ−1 tr Φ ≥ . lim inf µ→∞ tr Φ(q0 ) B (2 c)m−1 µ Theorem 1.5 is proved. ¯ has bounded geometry, there exist 5.2. Proof of Theorem 1.6. Since M ¯ ¯ satisfies constants c > 0 and R0 > 0 such that the sectional curvature of M 2 ¯ ¯ ¯ . We KM¯ ≤ c and the injectivity radius satisfies Rq ≥ R0 , for all q ∈ M 2 consider the constant functions K(t) = c and α(t) = κ ≥ 0, for all t. The function h(t) = 1c sin(c t), with t ∈ (0, πc ), is the maximal positive solution of (10). We take 0 < t0 ≤ 2πc the maximal positive number satisfying: h′ (t) = cos(c t) >

κ sin(c t) = α(t) h(t), c ′

2

(t) π for all 0 < t < t0 . Since 0 = α′ (t) ≥ − hh(t) 2 − K(t), for all t ∈ (0, 2c ], we obtain that µK,α = t0 . Let E be an end of M and λ : E → [0, ∞) a nonnegative C 1 function. The operator Φ(v) = λ(q)v, for all q ∈ E and v ∈ Tq M , satisfies |Φ(∇r)| = λ|∇r| ≤ λ = trmΦ , since |∇r| ≤ 1 and ¯0 } tr Φ = mλ. Thus Theorem 5.1 applies. Thus, for all 0 < µ < min{µK,α , R and q0 ∈ E such that Bµ (q0 ) ⊂ E, the following holds: Z λ ≥ λ(q0 ) Γ(µ), (43) Bµ (q0 )

Rµ m m−1 e−mκτ dµ > 0. where Γ(µ) = cm−1 0 sin(c τ ) Assume that lim sup x→∞ λ(x) > 0. Then there exists δ > 0 and a sequence x∈E (q1 , q2 , . . .) of points in E, with d(qk , x0 ) → ∞, where x0 is a fixed point of M , ¯ 0 }, after satisfying that λ(qk ) ≥ δ > 0, for all k. Fixed 0 < µ0 < min{µK,α , R a subsequence, we can assume that Bµ0 (qk ) ⊂ E and Bµ0 (qk ) ∩ Bµ0 (ql ) = R PN R ∅, for all k 6= l. Thus, by (43), we have that E λ ≥ k=1 Bµ0 (qk ) λ ≥ R N δ Γ(µ0 ), for all integer N ≥ 1. This implies that E λ = +∞. Theorem 1.6 is proved. Acknowledgement The authors thank Walcy Santos and Detang Zhou for helpful suggestions during the preparation of this article.

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References [1] Alencar, H., Santos, W. and Zhou, D., Curvature integral estimates for complete hypersurfaces, to appear in Illinois Math. Journal. [2] Barbosa, J. L. M. and Colares, A. G., Stability of hypersurfaces with constant r-mean curvature, Ann. Global Anal. Geom. 15 (1997), no. 3, 277 – 297. [3] Cao, H.-D., Shen, Y. and Zhu, S., The structure of stable minimal hypersurfaces in Rn+1 . Math. Res. Lett. 4 (1997), no. 5, 637 – 644. [4] Cheng, S.-T. and Yau, S.-T., Hypersurfaces with constant scalar curvature, Math. Ann. 255 (1977), 195 – 204. [5] Cheng, X., Cheung, L.-F. and Zhou, D., The structure of weakly stable constant mean curvature hypersurfaces. Tohoku Math. J. (2) 60 (2008), no. 1, 101 – 121. [6] do Carmo, M. P., Wang, Q., Xia, C., Complete submanifolds with bounded mean curvature in a Hadamard manifold. J. Geom. Phys. 60 (2010), no. 1, 142 – 154. [7] Federer, H., Geometric Measure Theory, Springer-Verlag New York Inc., New York, 1969, Die Grundlehren der mathematischen Wissenschaften, Band 153. [8] Federer, H., Curvature Measures, Trans. Amer. Math. Soc. 93 (1959), no. 3, 418 – 491. [9] Frensel, K. R., Stable complete surfaces with constant mean curvature, Bol. Soc. Brasil. Mat. (N.S.) 27 (1996), no. 2, 129 – 144. [10] Greene, R. E. and Wu, H., Function theory on manifolds which possess a pole, Lecture Notes in Mathematics, vol. 699, Springer, Berlin, 1979. [11] Reilly, R., Variational properties of functions of the mean curvatures for hypersurfaces in space forms, J. Differential Geometry 8 (1973), 465 – 477. [12] Rosenberg, H., Hypersurfaces of constant curvature in space forms, Bull. Sci. Math. 117 (1993), no. 2, 211 – 239. ´ tica, Universidade Federal de Alagoas, Maceio ´ , AL, Instituto de Matema CEP 57072-970, Brazil. E-mail address: [email protected] ´ tica, Universidade Federal do Rio de Janeiro, Rio de Instituto de Matema Janeiro, RJ, CEP 21945-970, Brazil. E-mail address: [email protected]